
diff -r 8ffa28fce271 -r b797c9a709d9 ChengsongTanPhdThesis/Chapters/Cubic.tex
--- a/ChengsongTanPhdThesis/Chapters/Cubic.tex	Sat Nov 12 21:34:40 2022 +0000
+++ b/ChengsongTanPhdThesis/Chapters/Cubic.tex	Thu Nov 17 23:13:57 2022 +0000
@@ -8,10 +8,14 @@
 \lstset{style=myScalastyle}
 
 
-This chapter is a ``miscellaneous''
-chapter which records various
-extensions to our $\blexersimp$'s formalisations.\\
-Firstly we present further improvements
+This chapter is a ``work-in-progress''
+chapter which records
+extensions to our $\blexersimp$.
+We intend to formalise this part, which
+we have not been able to finish due to time constraints of the PhD.
+Nevertheless, we outline the ideas we intend to use for the proof.
+
+We present further improvements
 made to our lexer algorithm $\blexersimp$.
 We devise a stronger simplification algorithm,
 called $\bsimpStrong$, which can prune away
@@ -35,13 +39,6 @@
 by exploring the connection between the internal
 data structure of our $\blexerStrong$ and
 Animirov's partial derivatives.\\
-Secondly, we extend our $\blexersimp$
-to support bounded repetitions ($r^{\{n\}}$).
-We update our formalisation of 
-the correctness and finiteness properties to
-include this new construct. 
-we can out-compete other verified lexers such as
-Verbatim++ on bounded regular expressions.
 %We also present the idempotency property proof
 %of $\bsimp$, which leverages the idempotency proof of $\rsimp$.
 %This reinforces our claim that the fixpoint construction
@@ -595,507 +592,6 @@
 %	SECTION 2
 %----------------------------------------------------------------------------------------
 
-\section{Bounded Repetitions}
-We have promised in chapter \ref{Introduction}
-that our lexing algorithm can potentially be extended
-to handle bounded repetitions
-in natural and elegant ways.
-Now we fulfill our promise by adding support for 
-the ``exactly-$n$-times'' bounded regular expression $r^{\{n\}}$.
-We add clauses in our derivatives-based lexing algorithms (with simplifications)
-introduced in chapter \ref{Bitcoded2}.
-
-\subsection{Augmented Definitions}
-There are a number of definitions that need to be augmented.
-The most notable one would be the POSIX rules for $r^{\{n\}}$:
-\begin{center}
-	\begin{mathpar}
-		\inferrule{\forall v \in vs_1. \vdash v:r \land 
-		|v| \neq []\\ \forall v \in vs_2. \vdash v:r \land |v| = []\\
-		\textit{length} \; (vs_1 @ vs_2) = n}{\textit{Stars} \;
-		(vs_1 @ vs_2) : r^{\{n\}} }
-	\end{mathpar}
-\end{center}
-As Ausaf had pointed out \cite{Ausaf},
-sometimes empty iterations have to be taken to get
-a match with exactly $n$ repetitions,
-and hence the $vs_2$ part.
-
-Another important definition would be the size:
-\begin{center}
-	\begin{tabular}{lcl}
-		$\llbracket r^{\{n\}} \rrbracket_r$ & $\dn$ & 
-		$\llbracket r \rrbracket_r + n$\\
-	\end{tabular}
-\end{center}
-\noindent
-Arguably we should use $\log \; n$ for the size because
-the number of digits increase logarithmically w.r.t $n$.
-For simplicity we choose to add the counter directly to the size.
-
-The derivative w.r.t a bounded regular expression
-is given as 
-\begin{center}
-	\begin{tabular}{lcl}
-		$r^{\{n\}} \backslash_r c$ & $\dn$ & 
-		$r\backslash_r c \cdot r^{\{n-1\}} \;\; \textit{if} \; n \geq 1$\\
-					   & & $\RZERO \;\quad \quad\quad \quad
-					   \textit{otherwise}$\\
-	\end{tabular}
-\end{center}
-\noindent
-For brevity, we sometimes use NTIMES to refer to bounded 
-regular expressions.
-The $\mkeps$ function clause for NTIMES would be
-\begin{center}
-	\begin{tabular}{lcl}
-		$\mkeps \; r^{\{n\}} $ & $\dn$ & $\Stars \;
-		(\textit{replicate} \; n\; (\mkeps \; r))$
-	\end{tabular}
-\end{center}
-\noindent
-The injection looks like
-\begin{center}
-	\begin{tabular}{lcl}
-		$\inj \; r^{\{n\}} \; c\; (\Seq \;v \; (\Stars \; vs)) $ & 
-		$\dn$ & $\Stars \;
-		((\inj \; r \;c \;v ) :: vs)$
-	\end{tabular}
-\end{center}
-\noindent
-
-
-\subsection{Proofs for the Augmented Lexing Algorithm}
-We need to maintain two proofs with the additional $r^{\{n\}}$
-construct: the 
-correctness proof in chapter \ref{Bitcoded2},
-and the finiteness proof in chapter \ref{Finite}.
-
-\subsubsection{Correctness Proof Augmentation}
-The correctness of $\textit{lexer}$ and $\textit{blexer}$ with bounded repetitions
-have been proven by Ausaf and Urban\cite{AusafDyckhoffUrban2016}.
-As they have commented, once the definitions are in place,
-the proofs given for the basic regular expressions will extend to
-bounded regular expressions, and there are no ``surprises''.
-We confirm this point because the correctness theorem would also
-extend without surprise to $\blexersimp$.
-The rewrite rules such as $\rightsquigarrow$, $\stackrel{s}{\rightsquigarrow}$ and so on
-do not need to be changed,
-and only a few lemmas such as lemma \ref{fltsPreserves} need to be adjusted to 
-add one more line which can be solved by sledgehammer 
-to solve the $r^{\{n\}}$ inductive case.
-
-
-\subsubsection{Finiteness Proof Augmentation}
-The bounded repetitions are
-very similar to stars, and therefore the treatment
-is similar, with minor changes to handle some slight complications
-when the counter reaches 0.
-The exponential growth is similar:
-\begin{center}
-	\begin{tabular}{ll}
-		$r^{\{n\}} $ & $\longrightarrow_{\backslash c}$\\
-		$(r\backslash c)  \cdot  
-		r^{\{n - 1\}}*$ & $\longrightarrow_{\backslash c'}$\\
-		\\
-		$r \backslash cc'  \cdot r^{\{n - 2\}}* + 
-		r \backslash c' \cdot r^{\{n - 1\}}*$ &
-		$\longrightarrow_{\backslash c''}$\\
-		\\
-		$(r_1 \backslash cc'c'' \cdot r^{\{n-3\}}* + 
-		r \backslash c''\cdot r^{\{n-1\}}) + 
-		(r \backslash c'c'' \cdot r^{\{n-2\}}* + 
-		r \backslash c'' \cdot r^{\{n-1\}}*)$ & 
-		$\longrightarrow_{\backslash c'''}$ \\
-		\\
-		$\ldots$\\
-	\end{tabular}
-\end{center}
-Again, we assume that $r\backslash c$, $r \backslash cc'$ and so on
-are all nullable.
-The flattened list of terms for $r^{\{n\}} \backslash_{rs} s$
-\begin{center}
-	$[r_1 \backslash cc'c'' \cdot r^{\{n-3\}}*,\;
-	r \backslash c''\cdot r^{\{n-1\}}, \; 
-	r \backslash c'c'' \cdot r^{\{n-2\}}*, \;
-	r \backslash c'' \cdot r^{\{n-1\}}*,\; \ldots ]$  
-\end{center}
-that comes from 
-\begin{center}
-		$(r_1 \backslash cc'c'' \cdot r^{\{n-3\}}* + 
-		r \backslash c''\cdot r^{\{n-1\}}) + 
-		(r \backslash c'c'' \cdot r^{\{n-2\}}* + 
-		r \backslash c'' \cdot r^{\{n-1\}}*)+ \ldots$ 
-\end{center}
-are made of sequences with different tails, where the counters
-might differ.
-The observation for maintaining the bound is that
-these counters never exceed $n$, the original
-counter. With the number of counters staying finite,
-$\rDistinct$ will deduplicate and keep the list finite.
-We introduce this idea as a lemma once we describe all
-the necessary helper functions.
-
-Similar to the star case, we want
-\begin{center}
-	$\rderssimp{r^{\{n\}}}{s} = \rsimp{\sum rs}$.
-\end{center}
-where $rs$
-shall be in the form of 
-$\map \; f \; Ss$, where $f$ is a function and
-$Ss$ a list of objects to act on.
-For star, the object's datatype is string.
-The list of strings $Ss$
-is generated using functions 
-$\starupdate$ and $\starupdates$.
-The function that takes a string and returns a regular expression
-is the anonymous function $
-(\lambda s'. \; r\backslash s' \cdot r^{\{m\}})$.
-In the NTIMES setting,
-the $\starupdate$ and $\starupdates$ functions are replaced by 
-$\textit{nupdate}$ and $\textit{nupdates}$:
-\begin{center}
-	\begin{tabular}{lcl}
-		$\nupdate \; c \; r \; [] $ & $\dn$ & $[]$\\
-		$\nupdate \; c \; r \; 
-		(\Some \; (s, \; n + 1) \; :: \; Ss)$ & $\dn$ & %\\
-						     $\textit{if} \; 
-						     (\rnullable \; (r \backslash_{rs} s))$ \\
-						     & & $\;\;\textit{then} 
-						     \;\; \Some \; (s @ [c], n + 1) :: \Some \; ([c], n) :: (
-						     \nupdate \; c \; r \; Ss)$ \\
-						     & & $\textit{else} \;\; \Some \; (s @ [c], n+1) :: (
-						     \nupdate \; c \; r \; Ss)$\\
-		$\nupdate \; c \; r \; (\textit{None} :: Ss)$ & $\dn$ & 
-		$(\None :: \nupdate  \; c \; r \; Ss)$\\
-							      & & \\
-	%\end{tabular}
-%\end{center}
-%\begin{center}
-	%\begin{tabular}{lcl}
-		$\nupdates \; [] \; r \; Ss$ & $\dn$ & $Ss$\\
-		$\nupdates \; (c :: cs) \; r \; Ss$ &  $\dn$ &  $\nupdates \; cs \; r \; (
-		\nupdate \; c \; r \; Ss)$
-	\end{tabular}
-\end{center}
-\noindent
-which take into account when a subterm
-\begin{center}
-	$r \backslash_s s \cdot r^{\{n\}}$
-\end{center}
-counter $n$
-is 0, and therefore expands to 
-\begin{center}
-$r \backslash_s (s@[c]) \cdot r^{\{n\}} \;+
-\; \ZERO$ 
-\end{center}
-after taking a derivative.
-The object now has type 
-\begin{center}
-$\textit{option} \;(\textit{string}, \textit{nat})$
-\end{center}
-and therefore the function for converting such an option into
-a regular expression term is called $\opterm$:
-
-\begin{center}
-	\begin{tabular}{lcl}
-	$\opterm \; r \; SN$ & $\dn$ & $\textit{case} \; SN\; of$\\
-				 & & $\;\;\Some \; (s, n) \Rightarrow 
-				 (r\backslash_{rs} s)\cdot r^{\{n\}}$\\
-				 & & $\;\;\None  \Rightarrow 
-				 \ZERO$\\
-	\end{tabular}
-\end{center}
-\noindent
-Put together, the list $\map \; f \; Ss$ is instantiated as
-\begin{center}
-	$\map \; (\opterm \; r) \; (\nupdates \; s \; r \;
-	[\Some \; ([c], n)])$.
-\end{center}
-For the closed form to be bounded, we would like
-simplification to be applied to each term in the list.
-Therefore we introduce some variants of $\opterm$,
-which help conveniently express the rewriting steps 
-needed in the closed form proof.
-\begin{center}
-	\begin{tabular}{lcl}
-	$\optermOsimp \; r \; SN$ & $\dn$ & $\textit{case} \; SN\; of$\\
-				 & & $\;\;\Some \; (s, n) \Rightarrow 
-				 \textit{rsimp} \; ((r\backslash_{rs} s)\cdot r^{\{n\}})$\\
-				 & & $\;\;\None  \Rightarrow 
-				 \ZERO$\\
-				 \\
-	$\optermosimp \; r \; SN$ & $\dn$ & $\textit{case} \; SN\; of$\\
-				 & & $\;\;\Some \; (s, n) \Rightarrow 
-				 (\textit{rsimp} \; (r\backslash_{rs} s)) 
-				 \cdot r^{\{n\}}$\\
-				 & & $\;\;\None  \Rightarrow 
-				 \ZERO$\\
-				 \\
-	$\optermsimp \; r \; SN$ & $\dn$ & $\textit{case} \; SN\; of$\\
-				 & & $\;\;\Some \; (s, n) \Rightarrow 
-				 (r\backslash_{rsimps} s)\cdot r^{\{n\}}$\\
-				 & & $\;\;\None  \Rightarrow 
-				 \ZERO$\\
-	\end{tabular}
-\end{center}
-
-
-For a list of 
-$\textit{option} \;(\textit{string}, \textit{nat})$ elements,
-we define the highest power for it recursively:
-\begin{center}
-	\begin{tabular}{lcl}
-		$\hpa \; [] \; n $ & $\dn$ & $n$\\
-		$\hpa \; (\None :: os) \; n $ &  $\dn$ &  $\hpa \; os \; n$\\
-		$\hpa \; (\Some \; (s, n) :: os) \; m$ & $\dn$ & 
-		$\hpa \;os \; (\textit{max} \; n\; m)$\\
-		\\
-		$\hpower \; rs $ & $\dn$ & $\hpa \; rs \; 0$\\
-	\end{tabular}
-\end{center}
-
-Now the intuition that an NTIMES regular expression's power
-does not increase can be easily expressed as
-\begin{lemma}\label{nupdatesMono2}
-	$\hpower \; (\nupdates \;s \; r \; [\Some \; ([c], n)]) \leq n$
-\end{lemma}
-\begin{proof}
-	Note that the power is non-increasing after a $\nupdate$ application:
-	\begin{center}
-		$\hpa \;\; (\nupdate \; c \; r \; Ss)\;\; m \leq 
-		 \hpa\; \; Ss \; m$.
-	 \end{center}
-	 This is also the case for $\nupdates$:
-	\begin{center}
-		$\hpa \;\; (\nupdates \; s \; r \; Ss)\;\; m \leq 
-		 \hpa\; \; Ss \; m$.
-	 \end{center}
-	 Therefore we have that
-	 \begin{center}
-		 $\hpower \;\; (\nupdates \; s \; r \; Ss) \leq
-		  \hpower \;\; Ss$
-	 \end{center}
-	 which leads to the lemma being proven.
-
- \end{proof}
-
-
-We also define the inductive rules for
-the shape of derivatives of the NTIMES regular expressions:\\[-3em]
-\begin{center}
-	\begin{mathpar}
-		\inferrule{\mbox{}}{\cbn \;\ZERO}
-
-		\inferrule{\mbox{}}{\cbn \; \; r_a \cdot (r^{\{n\}})}
-
-		\inferrule{\cbn \; r_1 \;\; \; \cbn \; r_2}{\cbn \; r_1 + r_2}
-
-		\inferrule{\cbn \; r}{\cbn \; r + \ZERO}
-	\end{mathpar}
-\end{center}
-\noindent
-A derivative of NTIMES fits into the shape described by $\cbn$:
-\begin{lemma}\label{ntimesDersCbn}
-	$\cbn \; ((r' \cdot r^{\{n\}}) \backslash_{rs} s)$ holds.
-\end{lemma}
-\begin{proof}
-	By a reverse induction on $s$.
-	For the inductive case, note that if $\cbn \; r$ holds,
-	then $\cbn \; (r\backslash_r c)$ holds.
-\end{proof}
-\noindent
-In addition, for $\cbn$-shaped regular expressioins, one can flatten
-them:
-\begin{lemma}\label{ntimesHfauPushin}
-	If $\cbn \; r$ holds, then $\hflataux{r \backslash_r c} = 
-	\textit{concat} \; (\map \; \hflataux{\map \; (\_\backslash_r c) \;
-	(\hflataux{r})})$
-\end{lemma}
-\begin{proof}
-	By an induction on the inductive cases of $\cbn$.
-\end{proof}
-\noindent
-This time we do not need to define the flattening functions for NTIMES only,
-because $\hflat{\_}$ and $\hflataux{\_}$ work on NTIMES already.
-\begin{lemma}\label{ntimesHfauInduct}
-$\hflataux{( (r\backslash_r c) \cdot r^{\{n\}}) \backslash_{rsimps} s} = 
- \map \; (\opterm \; r) \; (\nupdates \; s \; r \; [\Some \; ([c], n)])$
-\end{lemma}
-\begin{proof}
-	By a reverse induction on $s$.
-	The lemmas \ref{ntimesHfauPushin} and \ref{ntimesDersCbn} are used.
-\end{proof}
-\noindent
-We have a recursive property for NTIMES with $\nupdate$ 
-similar to that for STAR,
-and one for $\nupdates $ as well:
-\begin{lemma}\label{nupdateInduct1}
-	\mbox{}
-	\begin{itemize}
-		\item
-			\begin{center}
-	 $\textit{concat} \; (\map \; (\hflataux{\_} \circ (
-	\opterm \; r)) \; Ss) = \map \; (\opterm \; r) \; (\nupdate \;
-	c \; r \; Ss)$\\
-	\end{center}
-	holds.
-\item
-	\begin{center}
-	 $\textit{concat} \; (\map \; \hflataux{\_}\; 
-	\map \; (\_\backslash_r x) \;
-		(\map \; (\opterm \; r) \; (\nupdates \; xs \; r \; Ss)))$\\
-		$=$\\
-	$\map \; (\opterm \; r) \; (\nupdates \;(xs@[x]) \; r\;Ss)$ 
-	\end{center}
-	holds.
-	\end{itemize}
-\end{lemma}
-\begin{proof}
-	(i) is by an induction on $Ss$.
-	(ii) is by an induction on $xs$.
-\end{proof}
-\noindent
-The $\nString$ predicate is defined for conveniently
-expressing that there are no empty strings in the
-$\Some \;(s, n)$ elements generated by $\nupdate$:
-\begin{center}
-	\begin{tabular}{lcl}
-		$\nString \; \None$  & $\dn$ & $ \textit{true}$\\
-		$\nString \; (\Some \; ([], n))$ & $\dn$ & $ \textit{false}$\\
-		$\nString \; (\Some \; (c::s, n))$  & $\dn$ & $ \textit{true}$\\
-	\end{tabular}
-\end{center}
-\begin{lemma}\label{nupdatesNonempty}
-	If for all elements $o \in \textit{set} \; Ss$,
-	$\nString \; o$ holds, the we have that
-	for all elements $o' \in \textit{set} \; (\nupdates \; s \; r \; Ss)$,
-	$\nString \; o'$ holds.
-\end{lemma}
-\begin{proof}
-	By an induction on $s$, where $Ss$ is set to vary over all possible values.
-\end{proof}
-
-\noindent
-
-\begin{lemma}\label{ntimesClosedFormsSteps}
-	The following list of equalities or rewriting relations hold:\\
-	(i) $r^{\{n+1\}} \backslash_{rsimps} (c::s) = 
-	\textit{rsimp} \; (\sum (\map \; (\opterm \;r \;\_) \; (\nupdates \;
-	s \; r \; [\Some \; ([c], n)])))$\\
-	(ii)
-	\begin{center}
-	$\sum (\map \; (\opterm \; r) \; (\nupdates \; s \; r \; [
-	\Some \; ([c], n)]))$ \\ $ \sequal$\\
-	 $\sum (\map \; (\textit{rsimp} \circ (\opterm \; r))\; (\nupdates \;
-	 s\;r \; [\Some \; ([c], n)]))$\\
- 	\end{center}
-	(iii)
-	\begin{center}
-	$\sum \;(\map \; (\optermosimp \; r) \; (\nupdates \; s \; r\; [\Some \;
-	([c], n)]))$\\ 
-	$\sequal$\\
-	 $\sum \;(\map \; (\optermsimp r) \; (\nupdates \; s \; r \; [\Some \;
-	([c], n)])) $\\
-	\end{center}
-	(iv)
-	\begin{center}
-	$\sum \;(\map \; (\optermosimp \; r) \; (\nupdates \; s \; r\; [\Some \;
-	([c], n)])) $ \\ $\sequal$\\
-	 $\sum \;(\map \; (\optermOsimp r) \; (\nupdates \; s \; r \; [\Some \;
-	([c], n)])) $\\
-	\end{center}
-	(v)
-	\begin{center}
-	 $\sum \;(\map \; (\optermOsimp r) \; (\nupdates \; s \; r \; [\Some \;
-	 ([c], n)])) $ \\ $\sequal$\\
-	  $\sum \; (\map \; (\textit{rsimp} \circ (\opterm \; r)) \;
-	  (\nupdates \; s \; r \; [\Some \; ([c], n)]))$
-  	\end{center}
-\end{lemma}
-\begin{proof}
-	Routine.
-	(iii) and (iv) make use of the fact that all the strings $s$
-	inside $\Some \; (s, m)$ which are elements of the list
-	$\nupdates \; s\;r\;[\Some\; ([c], n)]$ are non-empty,
-	which is from lemma \ref{nupdatesNonempty}.
-	Once the string in $o = \Some \; (s, n)$ is 
-	nonempty, $\optermsimp \; r \;o$,
-	$\optermosimp \; r \; o$ and $\optermosimp \; \; o$ are guaranteed
-	to be equal.
-	(v) uses \ref{nupdateInduct1}.
-\end{proof}
-\noindent
-Now we are ready to present the closed form for NTIMES:
-\begin{theorem}\label{ntimesClosedForm}
-	The derivative of $r^{\{n+1\}}$ can be described as an alternative
-	containing a list
-	of terms:\\
-	$r^{\{n+1\}} \backslash_{rsimps} (c::s) = \textit{rsimp} \; (
-	\sum (\map \; (\optermsimp \; r) \; (\nupdates \; s \; r \;
-	[\Some \; ([c], n)])))$
-\end{theorem}
-\begin{proof}
-	By the rewriting steps described in lemma \ref{ntimesClosedFormsSteps}.
-\end{proof}
-\noindent
-The key observation for bounding this closed form
-is that the counter on $r^{\{n\}}$ will 
-only decrement during derivatives:
-\begin{lemma}\label{nupdatesNLeqN}
-	For an element $o$ in $\textit{set} \; (\nupdates \; s \; r \;
-	[\Some \; ([c], n)])$, either $o = \None$, or $o = \Some
-	\; (s', m)$ for some string $s'$ and number $m \leq n$.
-\end{lemma}
-\noindent
-The proof is routine and therefore omitted.
-This allows us to say what kind of terms
-are in the list $\textit{set} \; (\map \; (\optermsimp \; r) \; (
-\nupdates \; s \; r \; [\Some \; ([c], n)]))$:
-only $\ZERO_r$s or a sequence with the tail an $r^{\{m\}}$
-with a small $m$:
-\begin{lemma}\label{ntimesClosedFormListElemShape}
-	For any element $r'$ in $\textit{set} \; (\map \; (\optermsimp \; r) \; (
-	\nupdates \; s \; r \; [\Some \; ([c], n)]))$,
-	we have that $r'$ is either $\ZERO$ or $r \backslash_{rsimps} s' \cdot
-	r^{\{m\}}$ for some string $s'$ and number $m \leq n$.
-\end{lemma}
-\begin{proof}
-	Using lemma \ref{nupdatesNLeqN}.
-\end{proof}
-
-\begin{theorem}\label{ntimesClosedFormBounded}
-	Assuming that for any string $s$, $\llbracket r \backslash_{rsimps} s
-	\rrbracket_r \leq N$ holds, then we have that\\
-	$\llbracket r^{\{n+1\}} \backslash_{rsimps} s \rrbracket_r \leq
-	\textit{max} \; (c_N+1)* (N + \llbracket r^{\{n\}} \rrbracket+1)$,
-	where $c_N = \textit{card} \; (\textit{sizeNregex} \; (
-	N + \llbracket r^{\{n\}} \rrbracket_r+1))$.
-\end{theorem}
-\begin{proof}
-We have that for all regular expressions $r'$ in $\textit{set} \; (\map \; (\optermsimp \; r) \; (
-	\nupdates \; s \; r \; [\Some \; ([c], n)]))$,
-	$r'$'s size is less than or equal to $N + \llbracket r^{\{n\}} 
-	\rrbracket_r + 1$
-because $r'$ can only be either a $\ZERO$ or $r \backslash_{rsimps} s' \cdot
-r^{\{m\}}$ for some string $s'$ and number 
-$m \leq n$ (lemma \ref{ntimesClosedFormListElemShape}).
-In addition, we know that the list 
-$\map \; (\optermsimp \; r) \; (
-\nupdates \; s \; r \; [\Some \; ([c], n)])$'s size is at most
-$c_N = \textit{card} \; 
-(\sizeNregex \; ((N + \llbracket r^{\{n\}} \rrbracket) + 1))$.
-This gives us $\llbracket r \backslash_{rsimps} \;s \rrbracket_r
-\leq N * c_N$.
-\end{proof}
-
-We aim to formalise the correctness and size bound
-for constructs like $r^{\{\ldots n\}}$, $r^{\{n \ldots\}}$
-and so on, which is still work in progress.
-They should more or less follow the same recipe described in this section.
-Once we know about how to deal with them recursively using suitable auxiliary
-definitions, we are able to routinely establish the proofs.
-
 
 %The closed form for them looks like:
 %%\begin{center}




